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Non-normal approximation by Stein's method of exchangeable pairs with application to the Curie-Weiss model

Authors Catterjee, Sourav
Shao, Qi-Man View this author's profile
Issue Date 2010
Source The Annals of applied probability , v. 21, (2), 2011, p. 464-483
Summary Let (W, W') be an exchangeable pair. Assume that E(WW'\W) = g(W) + r(W), where g(W) is a dominated term and r(W) is negligible. Let G(t) = 0tg(s) ds and define p(t) = c1ec0G(t), where c0 is a properly chosen constant and c1 = 1 / −∞ec0G(t)dt. Let Y be a random variable with the probability density function p. It is proved that W converges to Y in distribution when the conditional second moment of (WW') given W satisfies a law of large numbers. A Berry–Esseen type bound is also given. We use this technique to obtain a Berry–Esseen error bound of order 1/sqrt(n) in the noncentral limit theorem for the magnetization in the Curie–Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli–Laplace Markov chain is also discussed.
ISSN 1050-5164
Language English
Format Article
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